subset of S is compact Let S be the set [0,1] and define a subset F of S to be closed if either it is finite or is equal to S Prove that this definition of closed set yields a topology for S Show that...

Let S be the set [0,1] and define a subset F of S to be

closed if either it is finite or is equal to S

Prove that this definition of closed set yields a topology

for S

Show that S with this topology is compact, but S is not a

Hausdorff space

Show that each subset of S is compact and that therefore there

are compact subsets of S that are not closed